Generation and numerical analysis (simulations in the static electric field, static magnetic field, dynamic magnetic field, and electro-magnetic field) of coupled equations, which consist of equations obtained by the finite element method and the boundary element method, require high-speed solution of large-scale simultaneous equations in the processing environment using parallel computers.
Increase in calculation speed requires the ordering of data, which is performed for the purpose of reducing the amount of communications between computers. Also, simultaneous equations based on the boundary element method use a large-scale matrix, and accordingly a region used for the preconditioning has to be limited. Further, the calculation time greatly depends upon the manner by which the size of the region is limited.
A method of directly solving large-scale simultaneous equations (direct method) has conventionally been adopted in the field of numerical analysis based on the combination of the finite element method and the boundary element method. The direct method imposes more loads for calculations than the iteration method, and requires immense calculation time.
A numerical analysis (simulations in the static electric field, static magnetic field, dynamic magnetic field, and electro-magnetic field) based on the combination of the finite element method and the boundary element method is advantageous because this analysis method allows consideration of movement of models while analyzing. It is generally known that simultaneous equations generated by the boundary element method present dense matrices having no zero component, which requires a long calculation time. Simultaneous equations having this characteristic can be solved by using the iteration method. However, these equations do not allow the preconditioning based on the incomplete Cholesky decomposition so that the convergence efficiency is low. Accordingly, the direct method is used for these equations. Further, even when the finite element method is used together, there are still dense regions in the matrices, and thus the direct method is used in the conventional method.
FIGS. 1A, 1B and 1C show types of matrices.
FIG. 1A shows a sparse matrix. A sparse matrix is a matrix in which coefficients outside of the vicinity of the on-diagonal component are zero. FIG. 1B shows a dense matrix. A dense matrix is a matrix that does not have a coefficient of zero. FIG. 1C shows a sparse-dense matrix. A sparse-dense matrix is a matrix in which some portions have a characteristic of the sparse matrix, and other portions have a characteristic of the dense matrix.
Patent Document 1 discloses a technique in which a preconditioning matrix for simultaneous linear equations is generated such that convergence of numerical solutions is enhanced in order to improve the calculation efficiency of parallel computers.    Patent Document 1:    Japanese Laid-open Patent Publication No. 5-81310